A related (or, actually, dual) problem is that of the stable finiteness of group rings (see [1]} for connections between surjunctivity, “linear surjunctivity” and stable finiteness).
Indeed, a ring \(R\)  is directly finite if \(xy=1\)  implies \(yx=1\)  for all \(x,y\in R\) . Furthermore, \(R\)  is stably finite if the ring of square \(k\times k\)  matrices \(\mathrm {Mat}_k(R)\)  is directly finite for all \(k\in \mathbb {N}_{\ge 1}\) . In other words, a ring \(R\)  is stably finite if, for each positive integer \(k\) , an endomorphism \(\phi \colon R^k\rightarrow R^k\)  is (split) surjective if and only if it is bijective.
A long standing open problem due to Kaplansky [2]} is to determine whether the group ring \(\mathbb {F}[G]\)  is stably finite for any field \(\mathbb {F}\) , we refer to this problem as the Stable Finiteness Conjecture.
In case the field \(\mathbb {F}\)  has characteristic 0, then the problem was solved in the positive by Kaplansky. There was no progress in the positive characteristic case until the year 2002, when Ara, O'Meara and Perera [3]} proved that a group algebra \(D[G]\)  is stably finite whenever \(G\)  is residually amenable and \(D\)  is any division ring. This last result was generalized by Elek and Szabó [4]}, that proved the same result for \(G\)  a sofic group (see also [1]} and [6]} for alternative proofs).
